We have $N\geq 9$ distinct positive reals numbers in range $[0,1)$ such that whenever we pick $8$ of them, there is another one (different from the first $8$) so that their sum is an integer. What are the possible values of $N$?
Clearly $N=9$ works, just take $9$ distinct reals adding to $1$. But I have not been able to figure out other values. However, we can notice that there are at least $\frac{\binom{N}{8}}{9}$ subsets of $9$ elements which have an integer sum, and this sum must be among $\{1,2,3,4,5,6,7,8\}$. So at least one of the numbers appears at least $\frac{\binom{N}{8}}{72}$ times. I get a feeling that this is impossible for large $N$.